Real-rooted atlas

A099040 → basis A013611 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(4 + 16\,x - 6\,n - 16\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(-56 - 240\,x - 256\,x^{2} + 36\,n + 80\,n x - 4\,n^{2}\right)\,P_{n-1}(x) + \left(-9 - 72\,x - 176\,x^{2} - 128\,x^{3} + 3\,n + 16\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
188
240128168
3160102415362720
45765504168961638439360
5195224832115200229376163840535200
663681011206282241900544278528015728646994400
72022438502429982721218969627197440314572801468006488928000

All 15 rows

Technical details
Node id
n-3ee2e71fbf42ce63 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f99aa997effff5f8

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      8;
  2:     40,     128;
  3:    160,    1024,     1536;
  4:    576,    5504,    16896,     16384;
  5:   1952,   24832,   115200,    229376,     163840;
  6:   6368,  101120,   628224,   1900544,    2785280,    1572864;
  7:  20224,  385024,  2998272,  12189696,   27197440,   31457280,   14680064;
  8:  62976, 1397760, 13086720,  66863104,  200867840,  353894400,  337641472,  134217728;
  9: 193152, 4896768, 53563392, 329678848, 1246658560, 2960130048, 4301258752, 3489660928, 1207959552;
%F Formula
The row polynomials satisfy (2*n^2-6*n+4-(16*n-16)*x)*P(n,x) = -(4*n^2-36*n+56-(80*n-240)*x+256*x^2)*P(n-1,x) + (3*n-9+(16*n-72)*x+(16*n-176)*x^2-128*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 36*n + 56 - (80*n - 240)*x + 256*x^2)*P[n - 1, x] + (3*n - 9 + (16*n - 72)*x + (16*n - 176)*x^2 - 128*x^3)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 - (16*n - 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 36*n + 56 - (80*n - 240)*x + 256*x^2)*v[i - 1] + (3*n - 9 + (16*n - 72)*x + (16*n - 176)*x^2 - 128*x^3)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 - (16*n - 16)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.